| L(s) = 1 | − 10.2·2-s − 23.7·4-s + 397.·5-s − 505.·7-s + 1.54e3·8-s − 4.05e3·10-s + 5.49e3·11-s + 5.26e3·13-s + 5.16e3·14-s − 1.27e4·16-s − 3.36e4·17-s + 4.18e4·19-s − 9.46e3·20-s − 5.60e4·22-s + 1.47e4·23-s + 8.00e4·25-s − 5.37e4·26-s + 1.20e4·28-s − 1.92e5·29-s − 2.22e5·31-s − 6.79e4·32-s + 3.43e5·34-s − 2.01e5·35-s + 3.19e5·37-s − 4.26e5·38-s + 6.16e5·40-s + 2.10e5·41-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.185·4-s + 1.42·5-s − 0.557·7-s + 1.07·8-s − 1.28·10-s + 1.24·11-s + 0.664·13-s + 0.502·14-s − 0.779·16-s − 1.66·17-s + 1.39·19-s − 0.264·20-s − 1.12·22-s + 0.252·23-s + 1.02·25-s − 0.599·26-s + 0.103·28-s − 1.46·29-s − 1.34·31-s − 0.366·32-s + 1.49·34-s − 0.792·35-s + 1.03·37-s − 1.26·38-s + 1.52·40-s + 0.475·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 10.2T + 128T^{2} \) |
| 5 | \( 1 - 397.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 505.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.18e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.47e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.92e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.19e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.97e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.46e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.50e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.283341900584875358096883684786, −8.888923333166004320968753676708, −7.50391250314874924395690265192, −6.55658594755225271060554917116, −5.79799244942850665667163141434, −4.60751795857316207242933197032, −3.42125784805198882037004044649, −1.90908144920685869130783744215, −1.28101272918109471174523389261, 0,
1.28101272918109471174523389261, 1.90908144920685869130783744215, 3.42125784805198882037004044649, 4.60751795857316207242933197032, 5.79799244942850665667163141434, 6.55658594755225271060554917116, 7.50391250314874924395690265192, 8.888923333166004320968753676708, 9.283341900584875358096883684786
